An index for set-valued maps in infinite-dimensional spaces

Abstract:

Previous fixed point indexes defined for a set-valued map in an infinite-dimensional space have required the values of this map to be convex sets. The corresponding assumption of this paper is that the values be (co-)acyclic sets, i.e., that the reduced Alexander cohomology group of each of these sets be trivial in each dimension. Other assumptions are that the space is locally convex and that the map is compact and upper semicontinuous with no fixed points on the boundary of its domain. The index is defined, proved to be homotopy invariant, and proved to vanish in case there are no fixed points. The main methods used are finite-dimensional approximation and the Vietoris-Begle mapping theorem.